reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th42:
  for P1,P2,S1 be FinSequence-membered set holds
   (P1\/P2)^S1 = P1^S1 \/ P2^S1
proof
  let P1,P2,S1 be FinSequence-membered set;
  thus (P1\/P2)^S1 c= P1^S1 \/ P2^S1
  proof
    let x be object such that
A1:   x in (P1\/P2)^S1;
    consider p, q be FinSequence such that
A2:   x = p^q & p in P1\/P2 & q in S1 by A1,POLNOT_1:def 2;
    p in P1 or p in P2 by A2,XBOOLE_0:def 3;
    then x in P1^S1 or x in P2^S1 by A2,POLNOT_1:def 2;
    hence thesis by XBOOLE_0:def 3;
  end;
A3: P1^S1 c= (P1\/P2)^S1
  proof
    let x be object such that
A4:   x in P1^S1;
    consider p, q be FinSequence such that
A5:   x = p^q & p in P1 & q in S1 by A4,POLNOT_1:def 2;
    p in P1\/P2 by A5,XBOOLE_0:def 3;
    hence thesis by A5,POLNOT_1:def 2;
  end;
  P2^S1 c= (P1\/P2)^S1
  proof
    let x be object such that
A6:   x in P2^S1;
    consider p, q be FinSequence such that
A7:   x = p^q & p in P2 & q in S1 by A6,POLNOT_1:def 2;
    p in P1\/P2 by A7,XBOOLE_0:def 3;
    hence thesis by A7,POLNOT_1:def 2;
  end;
  hence thesis by A3,XBOOLE_1:8;
end;
